Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

A norm is asymptotically uniformly convex (AUC) if $\overline\delta(t)=\inf_{\|x\|=1}\sup_{F}\inf_{y\in F,\|y\|=1}(\|x+ty\|-1)>0$ for all $t>0$ , $F$ ranging over finite-codimensional subspaces, and asymptotically midpoint uniformly convex (AMUC) if the same holds for the average of $\|x+ty\|$ and $\|x-ty\|$ . The countably branching diamonds $D_k$ replace each edge by countably many two-edge paths, $k$ times. Baudier, Causey, Dilworth, Kutzarova, Randrianarivony, Schlumprecht and Zhang (2017) proved that an equivalent AMUC norm prevents uniform bi-Lipschitz embeddings of the $D_k$ , and that for reflexive spaces with unconditional asymptotic structure, failure of AUC renormability forces uniform diamond embeddings; Swift and Perreau extended the latter under similar structure. Baudier and Lancien record the general question as Problem 39: if $X$ is reflexive and admits no equivalent AUC norm, must the countably branching diamonds embed into $X$ with uniformly bounded distortion?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Banach space theory: asymptotic geometry and metric embeddings
Posed by: F. P. Baudier and G. Lancien, Asymptotic and nonlinear geometries of Banach spaces and their interactions (arXiv 2512.00817, 2025; Cours Specialises 33, SMF 2026), Problem 39
Year posed: 2025
Years open: 1y
Solved: 2026-09-27
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 12 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: the dual $X=J^*$ of a countably branching segment-forest space with finite, unbounded component heights is infinite-dimensional, separable and reflexive, admits no equivalent AUC norm, and its given norm has averaged midpoint modulus $\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1$ ; every embedding of $D_k$ into $X$ has distortion at least $\sqrt{1+k/12}$ . So Problem 39 has a negative answer, and AMUC does not imply AUC renormability even for reflexive spaces (Baudier had shown the non-reflexive separation in 2026). Companions add midpoint-lens estimates, a sharper $D^2\ge1+k/4$ bound, exact moduli in a Daugavet subspace of $L_1$ , and further tree-potential and $L_1$ examples. Not shown: sharp distortion growth rates.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has seven manuscripts, all with Lean formalizations produced as part of the same release.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript against Problem 39 as the paper quotes it, and the Lean statement lean/ComparatorChallenges/ForestSpace.lean (OAI.ForestSpace.main_theorem), listed in formalization.yaml; not rebuilt here. The Lean statement covers the headline: the full dual of the specified segment-norm space is infinite-dimensional, separable and reflexive, no equivalent norm is AUC, the averaged midpoint modulus is at least $\sqrt{1+t^2/12}-1$ , and every embedding of the depth- $k$ diamond has distortion at least $\sqrt{1+k/12}$ . That is exactly a reflexive non-AUC-renormable space without uniform diamond embeddings. The six companions give further AMUC non-AUC examples, each with its own Comparator statement.

Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

What was actually shown

What the AI did

Verification

Sources

Discussion